Equation solutions {root, equation}| {zero, equation} are variable values that make equation true {solution, equation} {equation, solving}. Roots {null, equation} can be non-existent or zero. Equation-solution number equals equation degree. Two or more solutions can be equal {multiplicity, solution}.
order
Equations have highest exponent. Roots can be highest radical {order, radical} {radical, order}.
homogeneity
Equations can equal zero: f(x) = 0. Homogeneous equations have solutions.
polynomial
If polynomials have real-number coefficients, one factor is coefficient of highest-power term times product of polynomials of form (x - r) or (x^2 - (r + r')*x + r*(r')), where r is root and r' is root complex conjugate. Root complex conjugate is also a root.
complex number
If x and n are real, equation x^n = -1 solutions are complex numbers {roots of unity} {nth roots of unity}. For x^2 = -1, x = i. For x^3 = -1, x = i^(2/3). If w and z are complex, equation w^z = 1 solutions are log(1) = 0, 2 * pi * i, 4 * pi * i, 6 * pi * i, 8 * pi * i, ...
In unit circle, solutions are regular-polygon vertices, forming complex-number cyclic group Zn, a finite multiplicative group.
Mathematical Sciences>Algebra>Equation>Solution
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Date Modified: 2022.0224